Published by the Harvard Graduate School of Education
Harvard
Education
Letter
March/April 2007
Volume 23, Number 2
This article is reprinted
from the March/April 2007
issue of the Harvard
Education Letter.
interview with Sharon Griffin
“Doing the Critical Things First”
An aligned approach to preK and early elementary math
Along with learning the counting words, kids have to
be paying attention to the quantities in their world. The
first thing is to be able to describe quantities in terms of
polar differences: “This block tower is taller than that
block tower.” Taller, shorter; heavier, lighter; nearer, farther;
hotter, colder. If children are not paying attention to height
or weight or distance, then they’re not going to be able to
map numbers onto those dimensions very readily. In settings where there’s rich language, all of that vocabulary
What math skills do children need to have before they
mapping onto the world of quantity helps kids understand
start school?
the quantitative world in a global way. It
Children need to know the countmakes all these quantitative differences
ing words (“one, two, three …”) and to
salient, so that when they have a grasp
understand that this reliable sequence
of numbers, children can start mathemaMany teachers don’t
of words only gets meaning by being
tizing those quantities. All of this hapteach the link between
attached to quantities: “We’re going
pens very naturally in many homes, but
shopping and we’re going to get two
most of it is not taught in school (see
numbers and quantity.
boxes of cereal.” “You can have two
“Stages in the Development of Number
They assume all kids
cookies but not three.” To link these
Sense,” p. 5).
words to the quantities that give them
Children also need to become familhave it.
meaning is the crucial thing. Then they
iar with all the different ways numbers
can use the counting words by themare represented in our culture. Numselves without even needing to see the
bers can be represented as objects: Two
objects they are counting. That frees children up to do
means two things. Another way is in patterns, like the dot
math in their heads. If children establish the link between
patterns on dice or on playing cards. That’s already semianumber and quantity in the prekindergarten years, they
bstract, because the quantities are fixed. This makes it
will have a solid foundation for future learning.
harder to grasp that the five pattern is one more than the
four pattern—you can’t pick one dot up and put it back
What kinds of preK experiences do children need to
like you can with objects. Numbers can also refer to posidevelop these competencies?
tion. When you’re counting in a game like hopscotch, it’s
To begin with, they need to learn to say the counting
not that you’re adding one more object but you’re movwords in sequence—what I call the counting string. The
ing one more position in space from where you started. A
crucial thing about the counting words is learning to say
similar representation is vertical line representation, like
them in order. We can’t leave anything out, and we can’t
a thermometer, where numbers indicate quantity on a
say anything twice. It’s a sequential system.
continuous measurement scale. And the fifth way is dial
Sharon Griffin is an associate professor of education and
psychology at Clark University and author of the Number
Worlds curriculum for teaching number sense in the preK
and elementary years. In this interview with the Harvard
Education Letter, Griffin discusses what cognitive science
can teach us about aligning preK and early elementary curriculum and teaching methods with the natural development of children’s mathematical thinking.
Stages in the Development of Number Sense
Preschool: Counting words and quantities are not well linked
• Recite counting words (one to five or one to ten) in sequence
• Count with one-to-one correspondence
• Make global quantity comparisons (more, less; taller, shorter)
Kindergarten: Counting words and quantities become linked
• Know that small numbers mean a little and big numbers mean a lot
• Know that the next counting word in a sequence means one more
• Use numbers to compare quantities (two fewer cookies, six more steps)
• Can use numbers to make quantity determinations without using objects
First grade: Counting words and quantities are linked to symbols
• Recognize numerals and associate them with counting words
• Grasp the meaning of operational symbols (+, –, =)
• Form “number sentences” to express quantitative relationships (7 – 2 = 5)
bols like “4 + 2” on the board. They’re mapping counting words onto numerals and more
sophisticated algorithms, and kids don’t even
have a sense of what the counting words actually mean. By the time these children get to
second grade, it’s not surprising that many of
them hate math! My research shows that children who start kindergarten without this understanding and who do not receive remediation
fall farther and farther behind. But with proper,
explicit instruction that gives kids exposure
to the ways number is represented and allows
them to figure out the link between number
and quantity, they can catch up with and even
surpass their peers.
Once kindergartners have consolidated this
understanding, what should come next?
The business of first grade is to integrate the
Second grade: Children can link words to quantities along two scales
world of counting numbers and the world of
(e.g., tens and ones)
quantity with the world of formal symbols.
• Understand place value
That’s when you introduce numerals and plus
• Tell which two-digit number is bigger
signs and minus signs and equals signs and link
• Mentally solve two-digit addition and subtraction problems
them to the counting words, which children
• Solve problems involving hours and minutes, dollars and cents, weight
now associate with real quantities. But teachand distance
ers tend not to do this in the correct sequence.
They’ll put “5 + 3 = ” on the board and tell
Third grade: Children consolidate the understandings acquired in second grade
children, “Use your manipulatives to solve
• Apply understandings to broader range of concepts
this.” But kids may not even know what the
• Understand multiplication as a relationship among two groups
symbols mean, and therefore may not have any
(e.g., three cars with four wheels each)
idea how to represent them with objects. Let’s
Source: Adapted from S. Griffin, “The Development of Math Competence in the
let children solve problems with real quanti­Preschool and Early School Years,” in J.M. Royer, Mathematical Cognition.
ties—using cubes or weights or steps along a
line—let’s let them talk about what’s happening
before they start using numerals and symbols.
­representations, like a clock, in which quantities increase
“How many cubes do you have now? What happened—
as you move around the dial clockwise.
you got three more? Now how many do you have altogether? How did you figure that out?” Let them talk about
We know that many low-income children enter school
adding and subtracting, using the counting words and the
without the kinds of language-rich experiences that
middle-income children have typically had. What other language of quantity transactions (“I had five, and I gave
three away. Now I have two.”). Once that’s really solid,
gaps do you see at school entry and how do they affect
then introduce the symbols. But don’t do it backwards,
the development of children’s math skills?
which is our typical pattern.
More affluent children are exposed to different forms
This is the reason for the big push for communicaof representing numbers; lower-income kids much less
tion in the math classroom. We know that talking makes
so. Research shows that affluent kids are much more
likely to have board games in the home; they may also
have dominoes and playing cards. When they play Sorry,
for example, they learn that numbers indicate position
Editor’s Note
as they move around the board. To learn that numbers
This article is part of an ongoing series on the edumean position gives you a solid foundation for undercation of children from preK through grade 3, made
standing the ordinal value of numbers.
­possible through the support of the Foundation for
A lot of kids come into kindergarten with the link
Child Development. For additional information, visit
between number and quantity firmly established, usually
the ­Harvard Education Letter online resource, Focus on
the more affluent kids. A lot of lower-income kids don’t
Early Childhood Education, at www.hel-earlyed.org.
get this till they’re seven. It’s about a two-year delay. And
Special features include “Voices from the Field,” with
many teachers don’t teach the link between numbers and
comments on preK–3 math education from Herbert
quantity. They assume all kids have it. Even kindergarten
Ginsburg, Cathy Seeley, and Gene Maeroff.
teachers now will jump right in and write abstract sym-
Harvard Education Letter March/April 2007 Reprint
a huge difference. But it’s the one area that teachers are
inclined to cut because of time constraints. They want to
do a lot of drill and practice to prepare kids for the state
accountability tests, so they don’t give kids the chance to
make sense of the quantity transactions they are enacting, with or without manipulatives. That’s a huge tension
in elementary schools, and because of that it’s more and
more important that kids have that rich exposure in preschool and kindergarten, where there’s a little more flexibility in what teachers do.
Before they even start school, kids have two to three
years of rich experience in the world of language. We need
to build on that experience in school instead of forgetting
about it, and offer continuing opportunities to link formal
written symbols to this rich base of language (see “Developmental Principles for Early Math Instruction,” p. 6).
riculum standards called Focal Points, which narrows it
down to three areas for each grade level, starting in preK.
This is a huge step forward. Still, the amount of time
teachers have to teach math is often inadequate. There are
many interruptions in the classroom as children come and
go for special services, and the children who need quality math instruction the most are often the ones who are
pulled out of the classroom for other services.
Many math curricula emphasize “spiraling.” But if kids
don’t get a concept the first time it is taught, they are even
less likely to get it the second or third time, when it’s usually taught at a higher level. Spiraling prevents the teacher
from consolidating knowledge at each level before moving on, which is a basic principle of effective practice. For
instance, the function of third grade is to consolidate the
knowledge gained from preK through grade 2, in anticipation of the developmental gains that occur around grade 4.
In Asian schools, teachers take the time to teach fewer
concepts really well. They’ll show students something like
a balance beam, they’ll let them play with it, they’ll let
them predict and explain and argue among themselves.
In your ideal world, schools would take a developmentally sequenced approach …
… from preschool right up. Absolutely. Everything builds
on that early training. I believe very firmly that the only
way you can teach for understanding is to start
your instruction from where the child is. If you
don’t—you can teach kids tricks and rules and they
Developmental Principles for
might learn to use them, but they’re not going to
really understand them. If you don’t understand
Early Math Instruction
how these fundamental concepts are built, there’s
Teachers should:
no way that you can really interpret where a child
1.Build on children’s current knowledge
is and there’s no way you’ll have the resources to
2.Select learning objectives that are a natural next step
gear instruction to fit children’s needs.
Can you give an example?
At one time or another, almost all children count
on their fingers to solve single-digit math problems. To make them embarrassed about doing
this or to insist they do the math in their heads
deprives them of a wonderful tool for making sense
of numbers. Counting on their fingers doesn’t hinder cognitive development, it helps it. Children
will abandon this tool quite naturally when they
no longer need it.
Another example is telling time. Around grade
2, children reach a new stage of cognitive development in which they can start to grasp quantity along two dimensions—for instance, tens and
ones, hours and minutes, dollars and cents. At this
point it’s usually pretty easy for them to learn to
tell time. Yet state standards and curriculum guidelines often require students to master this skill in
first grade. These requirements aren’t appropriate to children’s level of understanding and expose
them to needless frustration.
You have said that today’s curricula are not well
aligned with the developmental sequence of
children’s mathematical thinking.
There are way too many math topics at every level.
In the fall of 2006, the National Council of Teachers of Mathematics came up with a new set of cur-
for children
3.Make sure children consolidate one level of
understanding before moving on to the next
4.Give children opportunities to use number concepts
in a broad range of contexts and to learn words for
describing quantity in each context (bigger, farther,
heavier, hotter)
Activities should:
•Expose children to the major ways numbers are
represented and talked about
•Provide opportunities to link quantities, counting
words, and symbols
•Provide visual and spatial analogs of number
representations for hands-on learning, such as
horizontal or vertical number lines children can use to
represent and visualize quantity transactions
•Capture children’s imaginations so knowledge is
embedded not only in their minds but in their hopes,
fears, and passions
•Provide opportunities to acquire computational fluency
as well as conceptual understanding
•Require the use of metacognitive processes (problemsolving, communication, reasoning) to help children
construct knowledge
For Further
Information

S. Griffin. “The Development of Math Competence
in the Preschool and Early
School Years: Cognitive
Foundations and Instructional Strategies.” In J.M.
Royer, ed., Mathematical
Cognition: Current Perspectives on Cognition,
Learning, and Instruction.
Charlotte, NC: Information
Age Publishing, 2002.
S. Griffin. “Fostering
the Development of
Whole-Number Sense:
Teaching Mathematics
in the Primary Grades.”
In M.S. Donovan and
J.D. Bransford, eds.,
How Students Learn:
History, Mathematics, and
Science in the Classroom.
Washington, DC: National
Academies Press, 2005.
S. Griffin. “Teaching
Number Sense: The
Cognitive Sciences Offer
Insights into How Young
Students Can Best Learn
Math.” Educational Leadership February 2004,
vol. 61, no. 5.
National Council of
Teachers of Mathematics. “Curriculum Focal
Points for Prekindergarten
through Grade 8 Mathematics.” Available online
at http://www.nctm.org/
focalpoints/default.asp
Number Worlds
http://clarku.edu/
numberworlds/
Source: Adapted from S. Griffin, “Fostering the Development of
Whole Number Sense,” in M.S. Donovan and J.D. Bransford,
How Students Learn.
Harvard Education Letter March/April 2007 Reprint Then they’ll test their predictions and at the end they’ll
say, “How can we describe this so we’ll know what to do
next time?” Then they start introducing the formal expressions. But we don’t have the time. We do it the quick-andeasy way for the kids who understand. We say, “You see
this symbol? This is what it means you do.”
What do teachers need to know in order to align their
math instruction with students’ cognitive development?
Preservice teachers come to my classes and they think
math is about numbers and rules for manipulating numbers. They don’t say, “Math is about quantity” or “Math
is a set of conceptual relationships between number and
quantity.” If you believe that math is about numbers and
the rules for manipulating them, of course the best thing
you can do is to teach kids the numbers and the rules as
early as possible. And when they say “numbers,” teachers tend not to think of spoken language. They think of
numerals, the written form of numbers. Of course children
Harvard Education Letter March/April 2007 Reprint
have to know the rules and the procedures, but they also
have to know when it’s appropriate to use them. Children
need to build that knowledge with rich experiences. The
point is not to teach all areas of mathematics from prekindergarten on up, but to cover them gradually over the
course of 18 years of schooling. You don’t have to do everything in every grade. Let’s do the critical things first and
do them well. n
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